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| Format: | Preprint |
| Veröffentlicht: |
2025
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| Schlagworte: | |
| Online-Zugang: | https://arxiv.org/abs/2511.21329 |
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| _version_ | 1866918218837262336 |
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| author | Chen, Chien-Hua |
| author_facet | Chen, Chien-Hua |
| contents | In this paper, we develop a view of self-isogenous modular polynomials and the $\mathfrak{l}$-cyclic isogeny graph for CM Drinfeld modules of arbitrary rank $r$. On the computational side, we give an explicit procedure to construct the modular polynomial $Φ_{J,\mathfrak{a}}(X,X)$ for Drinfeld modules of rank $r\geqslant 3$ with $\mathfrak{a}$ a prime ideal of $\mathbb{F}_q[T]$. When $\mathfrak{a}=(T)$, we provide an algorithm to compute $Φ_{J,\mathfrak{a}}(X,X)$; when $\mathfrak{a}=(T^2+T+1)$, we give an explicit degree bound on $Φ_{J,\mathfrak{a}}(X,X)$. On the structural side, we formulate a generalized $\mathfrak{l}$-cyclic volcano structure and prove that the generalized volcano appears in a component of the full $\mathfrak{l}$-cyclic isogeny graph for rank-$r$ Drinfeld modules with complex multiplication. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_21329 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | CM Drinfeld Modules, Self-isogenous Modular Polynomials, and Volcano Structure Chen, Chien-Hua Number Theory In this paper, we develop a view of self-isogenous modular polynomials and the $\mathfrak{l}$-cyclic isogeny graph for CM Drinfeld modules of arbitrary rank $r$. On the computational side, we give an explicit procedure to construct the modular polynomial $Φ_{J,\mathfrak{a}}(X,X)$ for Drinfeld modules of rank $r\geqslant 3$ with $\mathfrak{a}$ a prime ideal of $\mathbb{F}_q[T]$. When $\mathfrak{a}=(T)$, we provide an algorithm to compute $Φ_{J,\mathfrak{a}}(X,X)$; when $\mathfrak{a}=(T^2+T+1)$, we give an explicit degree bound on $Φ_{J,\mathfrak{a}}(X,X)$. On the structural side, we formulate a generalized $\mathfrak{l}$-cyclic volcano structure and prove that the generalized volcano appears in a component of the full $\mathfrak{l}$-cyclic isogeny graph for rank-$r$ Drinfeld modules with complex multiplication. |
| title | CM Drinfeld Modules, Self-isogenous Modular Polynomials, and Volcano Structure |
| topic | Number Theory |
| url | https://arxiv.org/abs/2511.21329 |